\section{1.7} 
\begin{frame}[allowframebreaks]{1.7. }

\vspace{-0.4cm}

{\color{red}1.7. PROPOSITION.} Let $M$ be a $\mathcal{D}_X$-module which is $\mathcal{O}_X$-coherent. 

Then it is locally free over $\mathcal{O}_X$.

This is a local problem. 

Fix $x \in X$. 

Given a local section $s$ of $M$ around $x$, let $\overline{s}$ be its image in the vector space $\overline{M}_x = M_x / \mathfrak{m}_x M$ over $\mathcal{O}_{X,x} / \mathfrak{m}_x = \mathbb{C}$. 

We may find finitely many $s_i$ ($1 \le i \le q$) such that the $\overline{s}_i$ form a vector space basis of $\overline{M}_x$. 

By the Nakayama lemma, they generate $M$ locally over $\mathcal{O}_X$. 

There remains to see the $s_i$ are linearly independent over $\mathcal{O}_X$. 

Assume we have a relation
\begin{equation}
\sum \phi_i \cdot s_i = 0
\end{equation}

Then we must have $\phi_i(x) = 0$, for $i = 1,\ldots,q$. 

Let $\nu$ be the minimum of the orders of the zeroes of the $\phi_i$ at $x$. 

It is $> 0$. 

To arrive at a contradiction, an induction on $\nu$ shows that it suffices to construct a non-trivial relation with a strictly smaller $\nu$.

Assume that $\phi_1$ say, has order $\nu$ at $0$. 

Using local coordinates, we can find $\partial \in \Theta_X$ such that $\partial \phi_1$ has a zero of order $> \nu$ at $x$. 

The relation $\mathcal{D}(\phi_1 s_1) = 0$ gives
\[
\mathcal{D}\partial \phi_1 \cdot s_1 + \mathcal{D}\phi_1 \cdot \partial s_1 = 0
\]

We may find regular functions $a_{ij}$ around $x$ such that
\[
\partial s_i = \sum_j a_{ij} s_j.
\]

We have then the relation
\[
\mathcal{D}s_1 (\partial \phi_1 + \mathcal{D}\phi_1 a_{1j} \phi_j) = 0
\]
where the coefficient of $s_1$ has at $x$ a zero of order $> \nu$.

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